Singularities of the Navier-Stokes and Euler equations

7 problems

• Regularity or blowup of axisymmetric Navier-Stokes flows with swirl

  Open

  Flows that are symmetric around an axis but also spin around it are the leading candidates for viscous blowup. Without spin they are known to stay smooth; with spin the question is open even in this restricted class.

• Finite-time blowup of 3D Euler from smooth data without boundaries

  Contested

  For an ideal fluid with no viscosity, it is unknown whether a smooth flow filling all of space can concentrate its rotation into infinite values in finite time. Blowup is proven when the fluid touches a wall or when the starting flow is slightly rough.

• Finite-time Navier-Stokes blowup driven by a smooth force

  Contested

  A machine-generated proof announced in September 2026 claims that a viscous fluid pushed by a smooth outside force can develop infinite velocity at one point in finite time. Whether the proof is correct decides two of the four official versions of the Clay problem.

• Uniqueness of Leray-Hopf weak solutions without forcing

  Open

  Weak solutions of the Navier-Stokes equations, which satisfy the equations only in an averaged sense, always exist, but it is unknown whether they are unique when no outside force acts. With a specially designed force, two different solutions from the same start are known.

• Global regularity or blowup of unforced 3D Navier-Stokes flows

  Open

  Prove that every smooth, finite-energy flow of a viscous incompressible fluid in 3D stays smooth forever, or find one that develops infinite velocity in finite time with no outside push. This is the unforced half of the Clay Millennium problem.

• Finite-time singularity of the inviscid surface quasi-geostrophic equation

  Open

  The surface quasi-geostrophic equation describes temperature carried by a flow it drives itself, and it behaves like a two-dimensional version of the 3D Euler equations. Whether a smooth temperature pattern can form an infinitely sharp front in finite time is unknown.

• Rigorous unstable self-similar singularities in incompressible fluid models

  Open

  Computer searches in 2025 found self-similar blowup solutions that occur only for finely tuned starting flows (unstable singularities) in simpler 2D fluid models. Turning these numerical profiles into proofs is open.